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On the whole line, the homogeneous one-dimensional wave equation with constant wave speed $c>0$ isHere $u$ is displacement in the string interpretation; $f$ is initial displacement and $g$ is initial velocity. Its two directions of travel giveThis is D'Alembert's formula, including both Cauchy data. The countertravelling decomposition of the one-dimensional wave equation identifies $F(x-ct)$ as the right-moving travelling wave and $G(x+ct)$ as the left-moving travelling wave. Their sum is permitted by the superposition principle. For $f\in C^2$ and $g\in C^1$, it is a classical solution. More generally, travelling waves with profiles $F$ and $G$ that are locally integrable functions solve the equation as distributions, with derivatives interpreted distributionally.^[ref]

The synthetic diagram uses the smooth bump function $b$ defined in D'Alembert's formula, with $f(x)=b(x)$ and $c=1$. Releasing this displacement from rest, $g=0$, gives $F=G=f/2$. Each emerging travelling wave has half the original peak. Launching the identical initial displacement solely to the right instead requires $g=-c{f}^{\prime }$ and gives $F=f$, $G=0$. At $t=1.5$, the released profiles have peaks $1/2$ at $x=\pm 1.5$, while the launched profile has peak $1$ at $x=1.5$. These are calculated model values. A purely left-moving launch requires $g=c{f}^{\prime }$. The initial velocity therefore selects the balance of directions; the initial shape alone does not.

Write $u(x,t)=U(\xi ,\eta )$, where $\xi =x-t$ and $\eta =x+t$. The chain rule givesFor a twice continuously differentiable classical solution, mixed partial derivatives commute, so the wave equation becomesHence $U_{\xi }$ depends only on $\xi$. Integrating first in $\xi$ yields the countertravelling decomposition of the one-dimensional wave equation:On a rectangular domain in $(\xi ,\eta )$ this follows immediately; on the full line for $t\ge 0$, the connected characteristic slices give the same representation. The two terms travel at wave speed one in opposite directions.

Impose the initial conditions. At $t=0$,Differentiate the first identity and combine:If $H(x)={\int }_{x_{\ast }}^{x}g(s)ds$, then $F=(f-H)/2+C$ and $G=(f+H)/2-C$. Substitution cancels $C$ and the arbitrary base point $x_{\ast }$, giving D'Alembert's formula:For example, $f\in C^2$ and $g\in C^1$ make this a classical solution of the wave equation satisfying both initial conditions. The same formula defines a weak solution for less regular data, including the continuous piecewise smooth data of part (b).

The printed arguments $x-t$ and $x+t$ assume unit wave speed. For the stated wave equation with general $c>0$, the correct arguments are $x-ct$ and $x+ct$, as in the paper's subsequent formula. Equivalently, use rescaled time $\tilde{t}=ct$ to obtain the printed unit-speed form.

Introduce coordinates $\xi =x-ct$, $\eta =x+ct$ along the characteristic lines for the one-dimensional wave equation, and write $u(x,t)=U(\xi ,\eta )$. The chain rule givesso the wave equation becomesThus $U_{\xi }$ is independent of $\eta$, and integration gives the countertravelling decomposition of the one-dimensional wave equationConversely, differentiating this expression twice verifies the wave equation for any sufficiently differentiable $F,G$.

The two initial conditions giveDifferentiating the first and solving these two equations for the derivatives givesFor a fixed reference point $x_0$, their antiderivatives can be writtenSubstituting the two travelling arguments cancels the constants and combines the integrals. This proves D'Alembert's formula with initial velocity:At $t=0$ the integral vanishes, giving $u=f$; differentiating by the fundamental theorem of calculus gives $u_t=g$ there. These checks also fix the sign and the factor $1/(2c)$.

For a classical solution take $F,G$ twice differentiable, so that the second derivatives exist. Direct use of the chain rule giveswith the two profile arguments as specified. Thus $4y_{tt}=25y_{xx}$.

To prove generality, introduce characteristic coordinates $\xi =2x+5t$, $\eta =2x-5t$, whose Jacobian determinant is $-20\ne 0$. For $y=U(\xi ,\eta )$ the differential operator isThe equation becomes $U_{\xi \eta }=0$. Integrating once in each variable yieldsThis is the countertravelling decomposition of the one-dimensional wave equation, with wave speed $5/2$. Once differentiability alone is not sufficient for a classical second-order equation; the displayed argument uses the required second derivatives.